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1615.vip:人教A版高中数学必修四 2.4.2 平面向量数量积的坐标表示、模、夹角 名师公开课市级获奖课件(16张)_图文

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复习引入 (1) a ? b ? a ? b cos ? ( 2) a ? a ? a 或 a ? 2 a ? a; a ?b a ?b . a ? b ? a ? b ? 0; cos ? ? 我们学过两向量的和与差可以转 化为它们相应的坐标来运算,那么怎 样用 a和b的坐标表示a ? b呢? 20:55 1 2.4.2 平面向量数量积 的坐标表示、模、夹角 平面向量数量积的坐标表示 j y轴 如图,是 i x轴上的单位向量,是 上的单位向量, 因为 a ? b ? a ? b cos? 所以 i ? i ?1 . j ? j ?1 . i ? j ? j ?i ? 0 . 20:55 y j o i 3 x 下面研究怎样用 a和b的坐标表示a ? b. 设两个非零向量 a =(x1,y1), b=(x2,y2),则 a ? x1 i ? y1 j 2 b ? x2 i ? y2 j 2 a ? b ? ( x1i ? y1 j) ? ( x2 i ? y2 j) ? x1 x2 i ? x1 y2 i ? j ? x2 y1 i ? j ? y1 y2 j 20:55 ? x1 x2 ? y1 y2 4 故两个向量的数量积等于它们对应 坐标的乘积的和。即 y A(x ,y ) 1 1 a ? b ? x1 x2 ? y1 y2 . B(x2,y2) b j a i o x 根据平面向量数量积的坐标表示,向 量的数量积的运算可转化为向量的坐标运 算。 20:55 5 向量的模和两点间距离公式: ?1?向量的模(长度公式):设a ? ( x, y), 则 ? a ? ? a ? a ? x ? y , 或 ? a ?? x ? y 2 2 2 2 2 ? 2? 设A? x1, y1 ?、B ? x2 , y2 ? , 则AB ? ? x2 ? x1 , y2 ? y1 ? ? AB ?? 20:55 ? x2 ? x1 ? ? ? y2 ? y1 ? 2 2 6 向量垂直和平行的坐标表示: a ? ? x1 , y1 ? , b ? ? x2 , y2 ? (a ? 0, b ? 0) (1)垂直: a ? b ? a ?b ? 0 ? x1 x2 ? y1 y2 ? 0 a / /b ? b ? ? a ? x1 y2 ? x2 y1 7 (2)平行: 20:55 例5 已知A(1,2),B(2,3),C(-2,5), 试判断?ABC的形状,并给出证明. 证明:? AB ? (2 ?1,3 ? 2) ? (1,1) AC ? (?2 ?1,5 ? 2) ? (?3,3) C(-2,5) y B(2,3) A(1,2) 0 8 ?AB? AC ? 1? (?3) ? 1? 3 ? 0 ? AB ? AC x ?三角形 ABC是直角三角形 . 20:55 练习:课本 107页 1,2 解:( 1 ) a ? b ? 2 ? (?2) ? 3? 4 ? 8 (2) ∵a ? b ? (0,7), a -b ? (4, -1 ) ? (a ? b ) ( ? a -b ) ? 0 - 7 ? -7 ( 3) ∵b ? c ? (?3,2) 20:55 ?a ? (b ? c) ? ?6 ? 6 ? 0 9 练习:课本 107页 1,2 (4)∵a ? b ? ( 0,7) ? (a ? b )? (a ? b ) ( ? a ?b ) ? 0 ? 49 ? 49 2 20:55 10 向量夹角公式的坐标表示: 设a ? ?x1 , y1 ?, b ? ?x2 , y2 ?, ?0 ? ? ? ? ? a与b夹角为?, a ? b ? x1 x2 ? y1 y2 cos ? ? 2 2 2 2 | a || b | x1 ? y1 . x2 ? y2 20:55 11 例6:设 a =( 5,-7),b=( -6,-4 ),求 a·b 及 a、b间的夹角θ(精确到1°). 解: a·b = 5×(-6)+(-7)×(-4)=-2 5×(-6)+(-7)×(-4) 52+(-7)2 × (-6)2+(-4)2 cosθ= ≈-0.0322413 θ≈-92° 20:55 12 2.设 a=(-1,1),b=(4,3),c=(5,-2). (1)求 a 与 b 的夹角的余弦值;(2)求 c 在 a 方向上的投影; (3)求λ1 和λ2,使 c=λ1a+λ2b. 全优60页变式训练 20:55 13 2.设 a=(-1,1),b=(4,3),c=(5,-2). (1)求 a 与 b 的夹角的余弦值;(2)求 c 在 a 方向上的投影; (3)求λ1 和λ2,使 c=λ1a+λ2b. 20:55 14 4.若a=(2,-2),则与a垂直的单位向量的坐 标为________. 全优93页 20:55 15 5.已知a,b为平面向量,且a=(4,3),2a+b=(3,18), 求a,b夹角的余弦值. 【解析】由已知得b=(3,18)-2a=(3,18)- (8,6)=(-5,12), 全优93页 20:55 16

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